Canonical sheaves at isolated canonical Gorenstein singularities

Abstract

It is well known that the Grauert-Riemenschneider canonical sheaf ${\mathcal{K}}_X$ of holomorphic square-integrable $n$-forms is a central tool in $L^2$-theory for the $\overline{\partial }$-operator on a singular complex space $X$ of pure dimension $n$. It was shown a few years ago that a comprehensive $L^2$-theory requires also the study of the sheaf ${\mathcal{K}}_X^s$ of holomorphic square-integrable $n$-forms with a Dirichlet boundary condition at the singular set of $X$. In the present paper, we describe and classify the behaviour of ${\mathcal{K}}_X^s$ in isolated canonical Gorenstein singularities, and give applications to the $L^2$-theory for the $\overline{\partial }$-operator on such spaces.